Langevin Simulation of Scalar Fields: Additive and Multiplicative Noises and Lattice Renormalization
N. C. Cassol-Seewald, R. L. S. Farias, E. S. Fraga, G. Krein, Rudnei O. Ramos
aa r X i v : . [ h e p - ph ] M a y Langevin Simulation of Scalar Fields:Additive and Multiplicative Noises and LatticeRenormalization
N. C. Cassol-Seewald a , R. L. S. Farias b , E. S. Fraga c , G. Krein a ,Rudnei O. Ramos d a Instituto de F´ısica Te´orica, Universidade Estadual Paulista,Rua Dr. Bento Teobaldo Ferraz, 271 - Bloco II, 01140-070 S˜ao Paulo, SP, Brazil b Departamento de Ciˆencias Naturais, Universidade Federal de S˜ao Jo˜ao Del Rei,36301-000 S˜ao Jo˜ao Del Rei, MG, Brazil c Instituto de F´ısica, Universidade Federal do Rio de Janeiro,21941-972 Rio de Janeiro, RJ, Brazil d Departamento de F´ısica Te´orica, Universidade do Estado do Rio de Janeiro,20550-013 Rio de Janeiro, RJ, Brazil
Abstract
We consider the Langevin lattice dynamics for a spontaneously broken λφ scalar field theory where both additive and multiplicative noise terms areincorporated. The lattice renormalization for the corresponding stochasticGinzburg-Landau-Langevin and the subtleties related to the multiplicativenoise are investigated. Keywords:
Langevin Dynamics, Lattice Renormalization, MultiplicativeNoise
PACS:
1. Introduction
The relevance of mathematical methods of quantum field theory for sta-tistical physics was recognized in the early seventies, particularly in the inves-tigation of equilibrium phase transitions and critical phenomena. Renormal-
Email addresses: [email protected] (N. C. Cassol-Seewald), [email protected] (R. L. S. Farias), [email protected] (E. S. Fraga), [email protected] (G. Krein), [email protected] (Rudnei O. Ramos)
Preprint submitted to Elsevier October 31, 2018 zation theory, originally linked with the removal of infinities in perturbativecalculations in quantum field theory, turned out to be a key element in theunderstanding of critical phenomena [1]. Functional integrals, Feynman dia-grams and loop expansions are an integral part of the mathematical methodsused presently in statistical physics – Ref. [2] is an excellent introductory texton concepts and methods of field theory in the quantum and statistical do-mains. Likewise, a large class of dynamic critical phenomena associated withtime-dependent fluctuations about equilibrium states, naturally describedin terms of stochastic partial differential equations [3, 4], can also be formu-lated in terms of functional integrals and therefore are equivalently describedby field theories [5]. Among the variety of phenomena associated with thedynamics of phase transitions, phase ordering [6] seems to be of particularimportance for the understanding of the time scales governing the equilibra-tion of systems driven out of equilibrium. The influence of the presence ofan environment on the dynamics of particles and fields is encoded “macro-scopically” in attributes that enter stochastic evolution equations, usually inthe form of dissipation and noise terms. Relevant time scales for differentstages of phase conversion can depend dramatically on the details of theseattributes. Noise terms, in particular, introduce difficulties in the numericalsimulation of the evolution equations; difficulties related to the well-knownRayleigh-Jeans catastrophe in classical field theory [7]. The present paper isconcerned with the use of field-theoretic renormalization methods to controlsuch catastrophe.A typical dynamical equation describing phase conversion is of the formof a Ginzburg-Landau-Langevin (GLL) equation [8]: τ ∂ ϕ∂t − ∇ ϕ + η ˙ ϕ + V ′ eff ( ϕ ) = ξ ( x , t ) , (1)where ϕ is a real scalar field, function of time and space variables, V ′ eff ( ϕ )is the field derivative of a Ginzburg-Landau effective potential and η , whichcan be seen as a response coefficient that defines time scales for the systemand encodes the intensity of dissipation, is usually taken to be a functionof temperature only, η = η ( T ). The function ξ ( x , t ) represents a stochastic(noise) force, assumed Gaussian and white, so that h ξ ( x , t ) i ξ = 0 , h ξ ( x , t ) ξ ( x ′ , t ′ ) i ξ = 2 ηT δ ( x − x ′ ) δ ( t − t ′ ) , (2)in conformity with the fluctuation-dissipation theorem. The condensate h ϕ i ξ h· · · i ξ means average over noiserealizations. The second order time derivative in Eq. (1) appears naturally inrelativistic field theories [9, 10] or when causality is incorporated via memoryfunctions [11, 12] in the otherwise purely diffusive, first-order evolution equa-tions [3, 8]. However, a more complete, microscopic field-theory descriptionof nonequilibrium dissipative dynamics [13] shows that the complete formfor the effective GLL equation of motion can lead to much more complicatedscenarios than the one described by Eq. (1), depending on the allowed inter-action terms involving ϕ . In general, there will be nonlocal (non-Markovian)dissipation and colored noise, as well as the possibility of field-dependent(multiplicative) noise terms ∼ ϕ ξ accompanied by density-dependent dis-sipation terms. Another typical example is provided by stochastic Gross-Pitaevskii (SGP) equations [14, 15] that incorporate thermal and quantumfluctuations of a Bose-Einstein condensate (BEC) on the traditional mean-field equation [16]. Such a stochastic equation can be derived from a micro-scopic inter-atomic Hamiltonian via the closed-time-path Schwinger-Keldysheffective action formalism [13]. In fact, on very general physical grounds, oneexpects that dissipation effects should depend on the local density ∼ ϕ ˙ ϕ and, accordingly, the noise term should contain a multiplicative piece ∼ ϕξ .The typical term coming from fluctuations in the equation of motion for ϕ will be a functional of the form [13] F [ ϕ ( x )] = ϕ ( x ) Z d x ′ ϕ ( x ′ ) K ( x, x ′ ) + Z d x ′ ϕ ( x ′ ) K ( x, x ′ ) , (3)where K ( x, x ′ ) and K ( x, x ′ ) are nonlocal kernels expressed in terms of re-tarded Green’s functions and whose explicit form depends on the detailednature of the interactions involving ϕ . Explicit treatments for these nonlocalkernels show that under appropriate conditions one is justified to express theeffective equation of motion for ϕ in a local form (see e.g. Ref. [17] and refer-ences therein). The existence of these additional terms in the GLL equationwill, of course, play an important role in the dynamics of the formation ofcondensates. For instance, it was shown that the effects of multiplicativenoise are rather significant in the Kibble-Zurek scenario of defect formationin one spatial dimension [18].Although in the literature there are many different approaches for study-ing the nonequilibrium dynamics in field theory [19, 20, 21], the use ofstochastic Langevin-like equations of motion is still a simpler and more directapproach in many different contexts in statistical physics and field theory in3eneral. For example, some of us have considered the effects of dissipationin the scenario of explosive spinodal decomposition: the rapid growth of un-stable modes following a quench into the two-phase region of the quantumchromodynamics (QCD) chiral transition in the simplest fashion [22]. Usinga phenomenological Langevin description inspired by microscopic nonequi-librium field theory results [10, 23, 24], the time evolution of the order pa-rameter in a chiral effective model [25] was investigated. Real-time (3 + 1)-dimensional lattice simulations for the behavior of the inhomogeneous chiralfields were performed, and it was shown that the effects of dissipation couldbe dramatic in spite of the very conservative assumptions that were made.Later, analogous but even stronger effects were obtained in the case of thedeconfining transition of SU (2) pure gauge theories using the same approach[26, 27]. Recent work in these directions by a different group can be foundin Refs. [28, 29].In the present paper we consider the nonequilibrium dynamics of the for-mation of a condensate in a spontaneously broken λϕ scalar field theorywithin an improved Langevin framework which includes the effects of multi-plicative noise and density-dependent dissipation terms. The correspondingstochastic GLL equation can be thought of as a generalization of the resultsof Ref. [10] to the case of broken symmetry. The time evolution for the for-mation of the condensate, under the influence of additive and multiplicativenoise terms, is solved numerically on a (3 + 1)-dimensional lattice. Particularattention is devoted to the renormalization of the stochastic GLL equationin order to obtain lattice-independent equilibrium results.The paper is organized as follows. In Section II the proper lattice renor-malization of the GLL, in order to achieve equilibrium solutions that areindependent of lattice spacing, is addressed. In Sec. III the question of timediscretization for a GLL with multiplicative noise is discussed. In Sec. IVwe show the results of our lattice simulations to study the behavior of thecondensate. Section VI contains our conclusions and perspectives.
2. Stochastic GLL equations
In our study, we consider an extended GLL equation, incorporating ad-ditive and multiplicative noise terms. The time evolution of the field ϕ ( x , t )at each point in space, which will determine the approach of the condensate h ϕ i to its equilibrium value will be dictated by the following equation:4 ϕ∂t − ∇ ϕ + (cid:0) η ϕ + η (cid:1) ∂ϕ∂t + V ′ eff ( ϕ ) = ξ ( x , t ) ϕ + ξ ( x , t ) , (4)where the dissipation coefficients η and η can be seen as response coeffi-cients that define time scales for the system and encode the intensity of dissi-pation. The functions ξ ( x , t ) and ξ ( x , t ) represent stochastic (noise) forces,assumed Gaussian and white, as in Eq. (2). The motivation for Eq. (4) stemslargely from explicit microscopic derivations of the effective equation of mo-tion for nonequilibrium quantum fields, where this type of equation emergesnaturally. We refer the interested reader to Refs. [10, 13] for more details. Analytic solutions of Eq. (4) are achievable only in very special situations(for the case of zero spatial dimensions, see e.g. Ref. [30]), like in a linear ap-proximation to V eff ( ϕ ), usually valid only at short times. Complete solutionsdescribing the evolution of the system to equilibrium can be obtained onlythrough extensive numerical simulations. In general, numerical simulationsare performed on a discrete spatial lattice of finite length under periodicboundary conditions. However, in performing lattice simulations of Eq. (4),one should be careful in preserving the lattice-spacing independence of theresults, especially when one is concerned with the behavior of the system inthe continuum limit. The equilibrium probability distribution for the fieldconfigurations φ that are solutions of Eq. (4) is P eq [ φ ] = e − S [ φ ] , where S [ φ ] isthe Euclidean action. The corresponding partition function is given by thepath integral Z [ φ ] = Z D φ e − S [ φ ] . (5)The calculation of expectation values and correlation functions of φ withthis partition function leads to ultraviolet divergences. In the presence ofthermal noise, short and long wavelength modes are mixed during the dy-namics, yielding an unphysical lattice spacing sensitivity. Such a latticespacing sensitivity is also present in the numerical simulation of SGP equa-tions [31, 32]. The issue of obtaining robust results, as well as the correctultraviolet behavior, in performing Langevin dynamics was discussed by sev-eral authors [33, 34, 35, 36, 37, 38]. 5he problem, which is not a priori evident in the Langevin formulation,is related to the well-known Rayleigh-Jeans ultraviolet catastrophe in clas-sical field theory [7]. The dynamics dictated by Eq. (4) is classical, and isill-defined for large momenta. These a priori lattice divergences can be elimi-nated by renormalizing the potential V eff through the addition of appropriatecounterterms (notice that these divergences are completely unrelated to theusual ones of the quantum theory). Since the divergent terms are all pertur-bative, one can identify the appropriate diagrams and subtract their resultcomputed within the classical theory . Since the theory in three-dimensionsis super-renormalizable, only a mass renormalization is required. We onlyrequire, then, a renormalization of the mass parameter m in V eff ( ϕ ). Us-ing such a renormalized potential in Eq. (4) leads to equilibrium solutions ϕ that are independent of the lattice spacing as we are going to explicitlyshow below. In practice, this lattice renormalization procedure correspondsto adding finite-temperature counterterms to the original potential V eff ( ϕ ),which guarantees the correct short-wavelength behavior of the discrete the-ory as was originally shown by the authors of Ref. [39] within the frameworkof dimensional reduction in a different context.In order to define the notation and the procedure to calculate the loopexpansion of the effective potential [40], we first consider the theory in thecontinuum and switch to the lattice when calculating the loop integrals. The calculation of the classical three-dimensional loop corrections to thepotential starts as usual by introducing a constant field ϕ through the trans-formation φ → φ + ϕ , (6)and considering the actionˆ S [ ϕ ; φ ] ≡ S [ ϕ + φ ] − S [ ϕ ] − Z d x φ ∂S [ φ ] ∂φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ = ϕ . (7)Next, the quadratic terms in ˆ S are collected in a “free” action ˆ S , and theremaining terms in an interacting action, ˆ S I . Then, the effective classicalfield potential is defined by the expression6 − βV V eff [ ϕ ] = e − βV V [ ϕ ] Z D φ e − ˆ S [ ϕ ; φ ] , (8)where V is the three-dimensional volume. From this expression, one obtainsfor V eff [ ϕ ] V eff [ ϕ ] = V [ ϕ ] − βV ln Z D φ e − ˆ S [ ϕ ; φ ] − βV ln h e − ˆ S I [ ϕ,φ ] i , (9)where h e − ˆ S I [ ϕ,φ ] i = R D φ e − ˆ S [ ϕ ; φ ] e − ˆ S I [ ϕ ; φ ] R D φ e − ˆ S [ ϕ ; φ ] . (10)The final step is the determination of ϕ through d V eff [ ϕ ] d ϕ = 0 . (11)For a bare potential of the form V = − m φ / λφ / S [ ϕ ; φ ]is given by ˆ S [ ϕ ; φ ] = β Z d x (cid:20) − φ ∇ φ + 12 m φ (cid:21) , (12)where m = − m + 12 λϕ . (13)Since ϕ is constant, the first functional integral in Eq. (9) can be easilyperformed in momentum space, with the result1 βV ln Z D φ e − ˆ S [ ϕ ; φ ] = − T Z k ln e G − [ ϕ ; k ] , (14)where e G − [ ϕ ; k ] is the inverse of the three-dimensional (classical field) prop-agator e G [ ϕ ; k ] = 1 k + m , (15)7nd R k ≡ R d k (2 π ) . The next divergent contribution comes from the two-loopcorrection to the mass, which will require ˆ S I ,ˆ S I [ ϕ ; φ ] = β Z d x (cid:18) − κφ + 14! λφ (cid:19) , (16)with κ = − λϕ . Expansion of e − ˆ S I in Eq. (9) gives the two-loop divergentcontribution, that we will denote as H [ ϕ ] and defined by1 βV ln h e − ˆ S I [ ϕ,φ ] i two − loop = T (cid:18) κ (cid:19) H [ ϕ ] , (17)where H [ ϕ ] = 16 V Z d xd y h φ ( x ) φ ( y ) i = Z k Z q e G [ ϕ ; k ] e G [ ϕ ; q ] e G [ ϕ ; ( k + q ) ] . (18)Now, the total divergent part of V eff [ ϕ ] is obtained from d V eff [ ϕ ] dϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) div . terms = λT I div [ ϕ ] − λ T H div [ ϕ ] , (19)with I [ ϕ ] = Z k e G [ ϕ ; k ] , (20)where I div [ ϕ ] and H div [ ϕ ] represent the divergent parts of I [ ϕ ] and H [ ϕ ].Notice that the derivatives of H [ ϕ ] with respect to ϕ lead to finite integralsand hence are irrelevant here. The evaluation of the divergent parts of I and H requires a regularization scheme. Since we simulate our GLL equationson a cubic lattice, we shall evaluate these divergent parts by calculating theeffective potential on the lattice. Here we consider the theory on a cubic lattice of volume V = L , with L = N a , where a is the lattice spacing and N is the number of lattice spacings.8he coordinates x i of the lattice sites are such that 0 ≤ x i ≤ a ( N − ∇ latt ≡ ∆, is defined as∆ φ ( x ) = 1 a X i =1 h φ ( x + a ˆ i ) − φ ( x ) + φ ( x − a ˆ i ) i , (21)where ˆ i is the unit cartesian vector indicating the three orthogonal directionsof the square lattice. We also impose periodic boundary conditions (PBC)on the fields, φ ( x + aN ˆ i ) = φ ( x ) , (22)and define the Fourier transform ˜ f ( k ) of a function f ( x ) on the lattice as˜ f ( k ) = a X x e − ik · x f ( x ) . (23)Because of the PBC, the allowed lattice momenta form the Brillouin zone B k i = 2 πaN n i , n i = 0 , , , · · · , N − . (24)The inverse transform is given by f ( x ) = 1 V X k ∈B e ik · x ˜ f ( k ) , (25)and the momentum summations over the Brillouin zone will be denoted by1 V X k ∈B ≡ Z k . (26)Let us now consider the derivation of the effective potential on the lattice.The lattice action is given by S latt [ φ ] = a X x (cid:18) − φ ∆ φ + V [ φ ] (cid:19) , (27)where V [ ϕ ] is the same as before. The derivation of the effective potentialfollows the same procedure as in the continuum, leading to expressions forthe one-loop and two-loop divergent contributions as in Eqs. (18) and (20),but with R k given by the sum over lattice momenta as indicated in Eq. (26).9hat remains to be determined is the lattice propagator corresponding to e G . The lattice propagator e G latt can be obtained from the quadratic actionˆ S [ ϕ ; φ ] on the latticeˆ S latt [ ϕ ; φ ] = a X x (cid:18) − φ ∆ φ + 12 m φ (cid:19) , (28)with m given by Eq. (13). The lattice propagator G latt [ ϕ ; x, y ] is the inverseof ( − ∆ + m ), i.e. a X y (cid:0) − ∆ + m (cid:1) x,y G latt [ ϕ ; x, y ] = 1 a δ x,y , (29)where δ x,y is the Kronecker delta. Since ϕ is a constant field, the solutionof this equation can be obtained using the Fourier transform of G latt [ ϕ ; x, y ],defined as G latt [ ϕ ; x, y ] = Z k e ik · x e G latt [ ϕ ; k ] . (30)Substituting this into Eq. (29), one obtains e G latt [ ϕ, k ] = a d ( ϕ ; n , n , n ) , (31)where d ( ϕ ; n , n , n ) = a " a − X i =1 (1 − cos ak i ) + m = X i =1 sin ( πn i /N ) + ( am/ . (32)One can then write I [ ϕ ] for the one-loop divergent term as I [ ϕ ] = 14 aN N X n i =0 d ( ϕ ; n , n , n ) , (33)and the double sum in H [ ϕ ] for the two-loop divergent term as10 [ ϕ ] = 164 N N − X n i ,m i =0 d ( ϕ ; n , n , n ) d ( ϕ ; m , m , m ) × d ( ϕ ; n + m , n + m , n + m ) . (34)The divergent parts of the sums above can be isolated in the limits of N → ∞ and a →
0. For example, in the one-loop term the sum in the limit of N → ∞ can be converted into an integral [39], I [ ϕ ] = 14 aπ Z π d x P i sin x i + ( am/ = 14 a Z ∞ dα e − α ( am ) / (cid:2) e − α/ I ( α/ (cid:3) , (35)where I is the modified Bessel function. In the limit of a →
0, the divergentpart of I [ ϕ ] is given by I div [ ϕ ] = Σ4 πa , (36)where Σ is a constant defined byΣ = 1 π Z + π/ − π/ d x P i sin x i ≃ . . (37)It is important to remark here that for T < T c it is crucial the use of Eq. (11)so that m < H div [ ϕ ] can also be isolated, with the result [39] H div [ ϕ ] = 116 π (cid:20) ln (cid:18) aM (cid:19) + ζ (cid:21) , (38)where M is the renormalization scale and ζ ≃ .
09 is another constant ap-pearing in the integrals being evaluated numerically.The renormalized mass in the classical field perturbative loop expansionis obtained as in the case of quantum field theory, i.e. by subtracting thedivergent parts of these graphs as indicated in Eq. (19):11 m φ → − (cid:0) m + δm (cid:1) φ ≡ − m R φ , (39)where the mass counterterm is given by δm = λT I div [ ϕ ] − λ T H div [ ϕ ] . (40)Therefore, we add the following finite-temperature counterterms to our orig-inal potential: V ct = (cid:26) − λ Σ8 π Ta + λ π T (cid:20) ln (cid:18) aM (cid:19) + ζ (cid:21)(cid:27) ϕ . (41)Notice from the above equation that the dependence on the mass scale M ofthe lattice counterterm is only logarithmic. So, it turns out that results areonly very weakly dependent on changes of the scale M . Of course, any changein the renormalization scale M can be compensated by corresponding changesin the renormalized parameters as also expected from the renormalizationgroup theory; here only a change in the coefficient of the term proportionalto ϕ is required, as it is clear from Eq. (41).
3. Multiplicative Noise and Time Discretization
The equation to be solved on the lattice is given by Eq. (4). We solvethe GLL equation on the lattice with periodic boundary conditions (PBC).When we insert the system in a box, ϕ acquires a discrete form ϕ nijk , where t = n ∆ t with n = 0 , , , . . . , x = ia , y = ja and z = ka , a being the latticespacing a = LN . Using PBC we have ϕ nN +1 jk = ϕ n jk ; ϕ niN +1 k = ϕ ni k ; ϕ nijN +1 = ϕ nij ,ϕ nN +1 N +1 k = ϕ n k ; ϕ nN +1 jN +1 = ϕ n j ; ϕ niN +1 N +1 = ϕ ni ,ϕ nN +1 N +1 N +1 = ϕ n ; ϕ n jk = ϕ nNjk ; ϕ ni k = ϕ niNk ,ϕ nij = ϕ nijN ; ϕ ni = ϕ niNN ; ϕ n j = ϕ nNjN ,ϕ n k = ϕ nNNk ; ϕ n = ϕ nNNN . (42)We write the Laplacian as 12 ϕ nijk = ∂ ϕ nijk ∂ x + ∂ ϕ nijk ∂ y + ∂ ϕ nijk ∂ z = 1 a (cid:20)(cid:18) ϕ ni +1 jk − ϕ nijk a (cid:19) − (cid:18) ϕ nijk − ϕ ni − jk a (cid:19) + (cid:18) ϕ nij +1 k − ϕ nijk a (cid:19) − (cid:18) ϕ nijk − ϕ nij − k a (cid:19) + (cid:18) ϕ nijk +1 − ϕ nijk a (cid:19) − (cid:18) ϕ nijk − ϕ nijk − a (cid:19)(cid:21) = 1 a (cid:2) ϕ ni +1 jk + ϕ nij +1 k + ϕ nijk +1 − ϕ nijk + ϕ ni − jk + ϕ nij − k + ϕ nijk − (cid:3) . (43)For simplicity, we introduce the compact notation ∇ ϕ nijk = ( Lϕ ) nijk . (44)In addition, we divide the time in n steps, t = n ∆ t , with n = 0 , , , . . . . Withrespect to the discretization of the time derivatives, we use the leapfrog ap-proximation method, where the algorithm is defined by the following iterationscheme: ∂ϕ n ∂t = ˙ ϕ n = 12 (cid:0) ˙ ϕ n +1 / + ˙ ϕ n − / (cid:1) , ˙ ϕ n +1 / = 1∆ t ( ϕ n +1 − ϕ n ) ,∂ ϕ n ∂t = ¨ ϕ n = 1∆ t (cid:0) ˙ ϕ n +1 / − ˙ ϕ n − / (cid:1) . (45)Now we rewrite (4) in terms of discrete the field ϕ nijk ∂ ϕ nijk ∂t − ∇ ϕ nijk + η (cid:0) ϕ nijk (cid:1) ∂ϕ nijk ∂t + η ∂ϕ nijk ∂t + V ′ eff (cid:0) ϕ nijk (cid:1) = ϕ nijk ξ + ξ . (46)Using the discretized quantities derived above we obtain13∆ t (cid:16) ˙ ϕ n +1 / ijk − ˙ ϕ n − / ijk (cid:17) = ( Lϕ ) nijk − η (cid:0) ϕ nijk (cid:1) (cid:16) ˙ ϕ n +1 / ijk + ˙ ϕ n − / ijk (cid:17) − η (cid:16) ˙ ϕ n +1 / ijk + ˙ ϕ n − / ijk (cid:17) − V ′ eff (cid:0) ϕ nijk (cid:1) + ϕ nijk ξ + ξ , (47)The equation that we solve iteratively is then (cid:26) h η (cid:0) ϕ nijk (cid:1) + η i ∆ t (cid:27) ˙ ϕ n +1 / ijk = (cid:26) − h η (cid:0) ϕ nijk (cid:1) + η i ∆ t (cid:27) ˙ ϕ n − / ijk + ( Lϕ ) nijk ∆ t − V ′ eff (cid:0) ϕ nijk (cid:1) ∆ t + ϕ nijk ξ ∆ t + ξ ∆ t . (48)In a compact notation we have˙ ϕ n +1 / ijk = 1Ξ h ˙ ϕ n − / ijk Θ + ( Lϕ ) nijk ∆ t − V ′ eff (cid:0) ϕ nijk (cid:1) ∆ t + ϕ nijk ξ ∆ t + ξ ∆ t (cid:3) , (49)where Ξ = 1 + 12 h η (cid:0) ϕ nijk (cid:1) + η i ∆ t , Θ = 1 − h η (cid:0) ϕ nijk (cid:1) + η i ∆ t . (50)To update the field we make: ϕ n +1 ijk = ϕ nijk + ∆ t ˙ ϕ n +1 / ijk . (51)On the lattice and with discretized time, the noise terms are modeled tosatisfy the discretized fluctuation-dissipation relation: h ξ i,n ξ j,n ′ i = 2 η i T δ i,j δ n,n ′ / ( a ∆ t ) , (52)with an amplitude that can then be written as14 i,n = r η i Ta ∆ t G i,n , (53)where G i,n is obtained from a zero-mean unit-variance Gaussian. Here, in-dices i = 1 , t ) / , instead of order ∆ t as the remaining terms. So,the multiplicative noise term needs to be re-expanded up the to next orderin ∆ t . This can be performed when one writes the discretized form for themultiplicative noise term using the Riemann formula, Z t +∆ tt ϕ ( x , t ′ ) ξ ( x , t ′ ) dt ′ = [(1 − α ) ϕ ( x , t ) + αϕ ( x , t + ∆ t )] χ ( x , t ) , (54)where 0 ≤ α ≤
1, with the Stratonovich prescription corresponding to α =1 /
2, while Ito’s corresponds to α = 0 [42]. In Eq. (54) χ ( x , t ) is a newGaussian stochastic process described by (in discretized form) h χ ,n χ ,n ′ i = 2 η T δ n,n ′ ∆ t/a . (55)Using Eq. (54) back in the leapfrog equation, Eq. (49), and re-expandingit to the next order in the multiplicative noise term, one finds that in theStratonovich interpretation there is an additional correction term to theleapfrog equation in Eq. (49) of order α ˙ ϕ n − / ijk χ ∆ t Θ / Ξ , which is alreadyan order (∆ t ) / higher than the remaining terms in that equation. We haveexplicitly checked in all our simulations that this is a negligible correction(which is consistent with the stochastic second order differential equationsin time discussed in [41]) for all our results, and so we have adopted Ito’sprescription, i.e. Eq. (49). Notice that this might not be the case would we15e working with first order in time derivative equations, in which case largecorrections due to the difference between Ito’s and Stratonovich’s interpreta-tions can arise [43]. For a discussion on different prescriptions in the contextof the relativistic Brownian motion, see e.g. Ref. [44].
4. Numerical Results
In this Section we present our results for the numerical simulations ofthe stochastic GLL equation in three spatial dimensions according to thediscretization method described in the previous section.Our interest is in the relaxation of a field to its equilibrium configura-tion. In order to analyze this behavior, we start by studying the dependenceof the solutions on the lattice spacing a = L/N and then, by introducingthe lattice counterterms discussed and derived in the previous section, weshow how lattice-independent results can be obtained in Langevin simula-tions using both standard additive noise and dissipation terms and also inits generalized form, which includes field-dependent (multiplicative) noiseand density-dependent dissipation. This is particularly important, since aswe have discussed in the Introduction, the effective equations of motion forbackground scalar fields turn out to be in general of the generalized Langevinform.All results that will be presented here refer to the time dependence of thevolume average of the noise-averaged order parameter, defined as h ϕ ( x, y, z, t ) i = 1 N X ijk ¯ ϕ nijk , (56)where ¯ ϕ nijk is the average over a large number N r of independent noise real-izations: ¯ ϕ nijk = 1 N r N r X r =1 ϕ nijk . (57)In all our numerical Langevin simulations, we consider N r between 20and 100. As usual, lattices with larger values of L require relatively lessrealizations over the noise. We have considered and tested different latticesizes to ensure the robustness of all numerical results.16 .1. The problem of lattice dependence in the generalized GLL approach As discussed in the previous section, the simulation of equations withnoise, being classical by nature, leads to the appearance of Rayleigh-Jeansultraviolet divergences at long times when simulating the equation on a dis-crete lattice. These divergences manifest themselves in the form of lattice-spacing dependence of the equilibrium solutions. We can show that by justconsidering the easiest nonequilibrium evolution, which is the one of relax-ation to the equilibrium state with initial conditions away from it. In Figs.1 and 2 we show the corresponding dynamics for the scalar field expectationvalue, h ϕ ( x, y, z, t ) i defined in Eqs. (56) and (57), for a symmetry-brokenGinzburg-Landau quartic potential, defined as V ( ϕ ) = λ (cid:0) T − T c (cid:1) ϕ λ ϕ , (58)for T < T c , with the critical temperature T c = 24 m /λ extracted from thefinite-temperature effective potential.The initial state for the field in the simulations for the broken phasewas taken around the inflexion (or spinodal) point of the finite temperature(Ginzburg-Landau) potential, ϕ infl , defined by d V ( ϕ, T ) dϕ (cid:12)(cid:12)(cid:12) ϕ = ϕ infl = 0 . (59)In Fig. 1 we show results for the standard Langevin equation, Eq. (1)with τ = 1, while Fig. 2 displays results for the generalized case, Eq. (4).The parameter values considered for the temperature and dissipation terms η and η , in units of m , were T /m = m η = η /m ≡
1, while thedimensionless quartic coupling constant was chosen to be λ = 0 .
25. Thescale M is taken as M/m = 1. These values suffice for our purposes of justdemonstrating the lattice dependence problem in Langevin simulations. InFigs. 1 and 2 the number of lattice points and the time stepsize were keptconstant, N = 64 and δt = 0 .
01 (in units of m ), respectively, while thelattice spacing, m δx ≡ a = L/N , was varied.It is clear that the solutions shown in Figs. 1 and 2 are not stable asthe lattice spacing is modified. As discussed previously, this problem can betraced to the fact that the equilibrium value of the quantity ¯ ϕ ijk gives theclassical average 17 m t ϕ / m m δ x = 1.0m δ x = 0.5m δ x = 0.25m δ x = 0.125 Figure 1: Solution of the standard GLL equation (1) using the leap frog algorithm fordifferent lattice spacings. ¯ ϕ ijk = R D ϕ ϕ ijk e − β V ( ϕ ) R D ϕ e − β V ( ϕ ) , (60)a divergent quantity. Thus, stable equilibrium solutions of the GLL equation,i.e., solutions not sensitive to lattice spacing, can only be obtained by theintroduction of the appropriate counterterms in the effective potential inorder to eliminate these divergences. These counterterms are the ones derivedin the previous section and given by Eq. (41).In Figs. 3 and 4 we present results of the simulations including thecounterterms. As we can see, equilibrium solutions that are independent oflattice spacing are obtained.For the standard GLL equation this was shown extensively in a series ofprevious papers [33, 34, 35, 36], but we are not aware of the same demon-stration for the case including multiplicative noise terms. From the resultsshown in Figs. 3 and 4 we can also immediately draw a couple of interestingconclusions. The first is that even though the counterterms used were cal-culated with an equilibrium partition function, thus ensuring lattice-spacingindependence only in equilibrium (large-time) situations, the results for shorttimes show only small lattice-spacing dependence. Second, we can notice that18
10 20 30 40 50 60 70 m t ϕ / m m δ x = 1.0m δ x = 0.5m δ x = 0.25m δ x = 0.125 Figure 2: Solution of the GLL equation with both additive and multiplicative noise anddissipation terms using the leap frog algorithm for different lattice spacings. The param-eters are the same as in Fig. 1. the relaxation time scales to the equilibrium state is about the same in allcases. Another observation we can make based on the results shown in Figs.3 and 4 is that the dynamics with multiplicative noise and dissipation termscan be quite different from that driven by additive noise only. In particular,notice from Fig. 4 that the relaxation time to the equilibrium state withthe generalized GLL equation is much longer than the one with just addi-tive noise, even though the magnitudes of the dissipation coefficients (in thedimensionless units in terms of m ) are of order one. One also notices thatthe difference in the dynamics of the two cases shown in Figs. 3 and 4, andin special the overdamped behavior seen in Fig. 4, is just a consequence ofthe parameters chosen. In particular, since in the multiplicative noise casethe dissipation term is proportional to the square of the amplitude of thefield, ϕ , and, for the parameters chosen, ϕ ∼ . −
5, this corresponds toa much more intense dissipation than the one for the case of additive noise,thus explaining the difference in the dynamics.
5. Conclusions
In this work we have studied several important aspects regarding thedynamics of a scalar field background configuration in broken phase. It19 m t ϕ / m m δ x = 1.0m δ x = 0.5m δ x = 0.25m δ x = 0.125 Figure 3: Solution of the standard GLL equation using the leap frog algorithm for differentlattice spacings and including the renormalization counterterms. has been long recognized that the effective evolution equation for the fieldcan be of a complicated form. From the analogy with standard Langevinequations for the study of the approach to equilibrium, the microscopicallyderived effective evolution equation allows for the presence of similar addi-tive noise and dissipation terms, but also for multiplicative (field-dependent)noise and dissipation contributions. Although equations of motion of thestandard Langevin form (with only additive noise and dissipation) have beenextensively studied in the literature, its generalized form, which includes themultiplicative noise and dissipation terms, still demands extensive studies.Here we have performed a number of numerical simulations with these equa-tions on a cubic lattice. We have also pointed out another issue frequentlyoverlooked in the literature: the necessity of adding lattice renormalizationcounterterms to cancel Rayleigh-Jeans divergences of the corresponding clas-sical theory in order to produce sensible results from Langevin simulations.We have shown that the same lattice counterterms that are required for thestandard Langevin simulations also work for the generalized Langevin equa-tions, producing lattice-independent equilibrium quantities, and also mini-mizing the dependence of the dynamics on the lattice parameters. One mustalso notice that the evaluation of correlation functions will also need appro-priate counterterms to render the results lattice independent. For example,20
10 20 30 40 50 60 70 m t ϕ / m m δ x = 1.0m δ x = 0.5m δ x = 0.25m δ x = 0.125 Figure 4: Solution of the GLL equation with both additive and multiplicative noise, anddissipation terms and including the renormalization counterterms. a quadratic correlation like h ϕ i will require a counterterm that can be ex-pressed in terms of I div , Eq. (36). A cubic correlation like h ϕ i will requirean additional counterterm that is given in terms of H div , Eq. (38). Higher-order correlations are expected to be free of divergences (i.e., will not requirecounterterms). That only these two terms, the ones which render the classi-cal effective potential finite, are necessary could be anticipated by recallingthat the effective potential is the generator of (zero-momentum) one-particleirreducible Green’s functions. Thus, higher-order Green’s functions shouldnot require other types of counterterms.One interesting and important problem that still remains, though be-yond the objectives set for the present work, is the study of the valid-ity of the approximation of transforming the complicated nonlocal (non-Markovian) equations of motion obtained through a microscopic derivation(via the Schwinger-Keldysh real-time formalism for the effective action) intothe local form used in the simulations performed for this study. This is anotoriously difficult problem due to the oscillatory nature of the nonlocalkernels appearing in the full effective equation of motion, which leads to un-controllable numerical behavior in simulations. Simulations and studies ofthe effects of the nonlocal terms in zero spatial dimensions [45, 46, 47] have21iven indications of the importance of the full non-Markovian dynamics com-pared with their local approximation. Implementing the space dependenceon the non-local kernels and a full simulation of the non-Markovian dynamicsis the subject of a future work that is expected to complement the presentinvestigation.
6. Acknowledgements
E.S.F. would like to thank T. Kodama, T. Koide, A. J. Mizher and L.F. Palhares for discussions on related matters. This work was partially sup-ported by CAPES, CNPq, FAPERJ, FAPESP, FAPEMIG and FUJB/UFRJ(Brazilian Agencies).
References [1] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, OxfordUniversity Press, Oxford, 2002.[2] M. Le Bellac, Quantum and statistical field theory, Oxford UniversityPress, Oxford, 1991.[3] P. C. Hohenberg, B. I. Halperin, Theory of Dynamic Critical Phenom-ena, Review of Modern Physics 49 (1977) 435-479.[4] N. G. van Kampen, Stochastic processes in physics and chemistry, 2ndEdition, North-Holland, Amsterdam, 1992. H. S. Wio, An introductionto stochastic processes and nonequilibrium statistical physics, Series onAdvances in Statistical Mechanics, Volume 10, World Scientific, Singa-pore, 1994.[5] P. C. Martin, E. D. Siggia, H. A. Rose, Statistical Dynamics of ClassicalSystems, Physical Review A 8 (1973) 423-437.[6] K. Kawasaki, Phase Transitions and Critical Phenomena, Volume 2,Editors C.Domb, M. S. Green, Academic, New York, 1976.[7] G. Parisi, Statistical Field Theory, Addison-Wesley, New York, 1988.[8] A. Onuki, Phase Transition Dynamics, Cambridge University Press,Cambridge, 2002. 229] M. Morikawa, Classical Fluctuations In Dissipative Quantum Systems,Physical Review D 33 (1986) 3607-3612.[10] M. Gleiser, R. O. Ramos, Microphysical approach to nonequilibriumdynamics of quantum fields, Physical Review D 50 (1994) 2441-2455.A. Berera, M. Gleiser, R. O. Ramos, Strong dissipative behavior inquantum field theory, Physical Review D 58 (1998) 123508.[11] T. Koide, G. Krein, R. O. Ramos, Incorporating memory effects in phaseseparation processes, Physics Letters B 636 (2006) 96-100.[12] N. C. Cassol-Seewald, M. I. M. Copetti, G. Krein, Numerical approx-imation of the Ginzburg-Landau equation with memory effects in thedynamics of phase transitions, Computer Physics Communication 179(2008) 297-309.[13] E. Calzetta, B.-L. Hu, Nonequilibrium Quantum Field Theory, Cam-bridge University Press, Cambridge, 2008.[14] E. Calzetta, B-L. Hu, E. Verdaguer, Stochastic Gross-Pitaievskii equa-tion for BEC via coarse-grained effective action, International Journalof Modern Physics B 21 (2007) 4239-4247.[15] H. T. C. Stoof, Journal of Low Temperature Physics 114 (1999) 11-108, H.T.C. Stoof, M. J. Bijlsma, Journal of Low Temperature Physics124 (2001) 431-442, C.W. Gardiner, M. J. Davis, The stochastic Gross-Pitaevskii equation: II, Journal of Physics B 36 (2003) 4731-4753.[16] E. P.Gross, Structure of a quantized vortex in boson systems, NuovoCimento 20 (1961) 454-457. L. P. Pitaevskii, Vortex Lines in an Imper-fect Bose Gas, Journal of experimental and theoretical physics of theAcademy of Sciences of the USSR 13 (1961) 451-454.[17] A. Berera, I. G. Moss, R. O. Ramos, Local approximations for effectivescalar field equations of motion, Physical Review D 76 (2007) 083520.[18] N. D. Antunes, P. Gandra, R. J. Rivers, The effects of multiplicativenoise in relativistic phase transitions, Physical Review D 71 (2005)105006. 2319] G. Aarts, J. Smit, Classical approximation for time dependent quan-tum field theory: Diagrammatic analysis for hot scalar fields, NuclearPhysics B 511 (1998) 451-478. G. Aarts, G. F. Bonini, C. Wetterich, OnThermalization in classical scalar field theory, Nuclear Physics B 587(2000) 403-418.[20] J. Berges, Controlled nonperturbative dynamics of quantum fields outof equilibrium, Nuclear Physics A 699 (2002) 847-886.[21] C. Destri, H. J. de Vega, Ultraviolet cascade in the thermalization of theclassical phi**4 theory in 3+1 dimensions, Physical Review D 73 (2006)025014.[22] E. S. Fraga, G. Krein, Can dissipation prevent explosive decompositionin high-energy heavy ion collisions?, Physics Letters B 614 (2005) 181-186.[23] C. Greiner, B. Muller, Classical Fields Near Thermal Equilibrium, Phys-ical Review D 55 (1997) 1026-1046.[24] D. H. Rischke, Forming disoriented chiral condensates through fluctua-tions, Physical Review C 58 (1998) 2331-2357.[25] O. Scavenius, A. Dumitru, E. S. Fraga, J. T. Lenaghan, A. D. Jackson,First order chiral phase transition in high-energy collisions: Can nucle-ation prevent spinodal decomposition?, Physical Review D 63 (2001)116003.[26] E. S. Fraga, T. Kodama, G. Krein, A. J. Mizher, L. F. Palhares, Dissi-pation and memory effects in pure glue deconfinement, Nuclear PhysicsA 785 (2007) 138-141.[27] E. S. Fraga, G. Krein, A. J. Mizher, Langevin dynamics of the pureSU(2) deconfining transition, Physical Review D 76 (2007) 034501.[28] M. Nahrgang, S. Leupold, C. Herold, M. Bleicher, Nonequilibrium chiralfluid dynamics including dissipation and noise, Physical Review C 84(2011) 024912.[29] M. Nahrgang, M. Bleicher, The QCD Phase Diagram in Chiral FluidDynamics, Acta Physica Polonica B Proceedings Supplement 4 (2011)609-614. 2430] R. L. S. Farias, R. O. Ramos, L. A. da Silva, Numerical Solutions for non-Markovian Stochastic Equations of Motion, Computer Physics Commu-nication 180 (2009) 574-579.[31] S. P. Cockburn, N. P. Proukakis, The stochastic Gross-Pitaevskii equa-tion and some applications, Laser Physics 19 (2009) 558-570.[32] S. P. Cockburn, A. Negretti, N. P. Proukakis, C. Henkel, A comparisonbetween microscopic methods for finite temperature Bose gases PhysicalReview A 83 (2011) 043619.[33] M. Gleiser and H. -R. Muller, How to count kinks: From the contin-uum to the lattice and back, Phys. Lett. B422